July 31, 2014

Poorly Executed Mnemonics Definitely Addle Students

If you've read my past two posts, you know by now that PEMDAS* (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is one of my most hated mnemonics. There are two main reasons:

It's misleading.

It's unnecessary.

First of all, why is PEMDAS misleading?

Let's start with the "P" (Parentheses). People claim that PEMDAS is "the order of operations." This is already problematic because parentheses aren't really a mathematical operation.** Operations do things. Parentheses don't actually do anything - they just group things together. This distinction may seem like more of a technicality, but it actually brings to light the main issue: parentheses aren't the important thing, but the idea of grouping in general. There's lots of ways to group expressions.

You can group expressions using a fraction bar. \[\frac{1+2}{3+4}\]

You can group expressions under a radical. \[\sqrt{3^2+4^2}\]

You can even group expressions inside an exponent! \[2^{4+1}\](How are you supposed to "do" exponents before addition if there's addition in the exponent and you don't have parentheses to tell you what to do?)

So in terms of grouping, PEMDAS is at best incomplete.

Though the "E" (Exponents) is pretty much unambiguous, the entire rest of the mnemonic causes problems. By putting the "MDAS" in linear order, a number of students get the idea that all Multiplication should be done before any Division, and that all Addition should be done before any Subtraction. Thus you get students who will make the following mistakes:\[4-1+2\\=4-3\\=1(?!)\]\[6\div2\times3\\=6\div6\\=1(?!)\]
What's even scarier is that PEMDAS has become so ingrained in our math education culture that some teachers actually teach it this blatantly incorrect way. Don't believe me? Take a look at this video from TED-Ed:

Try to tell me that doesn't lead you to believe that MDAS is done in linear order. And we wonder why kids have trouble.

Of course, many teachers are careful to explain how the order of operations are supposed to work - that multiplication and division (which is just multiplication by the multiplicative inverse) are done in order from left to right as they appear. Likewise, subtraction is just addition by the additive inverse, so addition and subtraction are done left to right as well. Some teachers write the mnemonic as PE(MD)(AS) or in some other sort of arrangement to emphasize this fact. Others further extend the ridiculous mnemonic-for-a-mnemonic to say "Please Excuse My Dear Aunt Sally ... and Let her Rest".

But now why is PEMDAS unnecessary?

There is a way to bypass all of this mnemonic madness and teach the order of operations in a way that actually makes sense. How?

By teaching why it works that way.

When I've asked fellow teachers why order of operations the way it is, those who have been able to answer often gave something to the effect of "well, we needed to decide on some kind of convention to deal with possible ambiguity, so we decided on what we have today." This is half correct - getting rid of ambiguity is very much important. But it wasn't an arbitrary decision. It's not like we could have just as equally decided that addition and subtraction come first, then multiplication and division, and then exponents. There's a very good reason that the operations fall naturally in the order that they do.

Think back to elementary school when all you knew about was addition and subtraction. Eventually you ran into expressions that looked like this:\[3+3+3+3+3+3+3\]You didn't want to write so many 3's, so you were introduced to a shorthand to write this expression. Since there were seven 3's being added together, you learned you could instead write:\[3\times7\]Thus you learned that multiplication is repeated addition.

Fast forward a few years, when you had multiplication and division under your belt. Now you saw expressions like this instead:\[4\times4\times4\times4\times4\]Again, you were introduced to a shorthand to keep from having to write all those 4's. Since there were five 4's being multiplied together, you wrote:\[4^5\]Thus you learned that exponentiation is repeated multiplication.

Now we come to an expression like this.\[5^2+4\times3\]What do we do first? Well, remembering that exponentiation is repeated multiplication, we rewrite our exponent to say what it really means.\[5\times5+4\times3\]Next, remembering that multiplication is repeated addition, we rewrite our multiplication in even more basic terms.\[5+5+5+5+5+4+4+4\]Now the expression is a cinch to evaluate - anyone can add! The value just comes out to 37. But, more remarkably, what we've just done is uncovered the reason why the order of operations is as it is:The most compact shorthand is evaluated first.

Once students understand this, they won't need to actually write out the additions explicitly - they'll just evaluate things in the order they should be handled. But they'll know why to do it.
With this in mind, I propose a better way to teach order of operations. Students need only remember two things.

Pay attention to grouping.

Shorthand comes first.

If we must use an acronym, instead of PEMDAS, let's use something like ... say ... GEMA. (Grouping, Exponentiation, Multiplication, Addition.) But if we do use GEMA or something similar, we shouldn't deprive students of the understanding that comes from knowing why the order of operations works.

* In other countries, variations on PEMDAS are used, such as BODMAS or BIDMAS. The "B" stands for Brackets, another word for parentheses, and "O" and "I" stand for Orders and Indices, respectively, which are both alternate words for exponents. Note that in BODMAS and BIDMAS, the "D" and "M" are interchanged - think about how much confusion that could cause for students!** Thanks to Quintopia for pointing this out: Parentheses in the context of computer science CAN in fact be thought of as operators which let the computer know to make a call to a subroutine. It would be even harder to make this into an acronym ... unless it's a recursive acronym in which the G stands for GEMA, similar to how WINE stands for WINE Is Not an Emulator!

10 comments
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Parentheses are call-to-subroutine operators. You can't use non-bracket grouping methods in ASCII, so you need parentheses to say when we need to perform subsequent operations on a new stack before inserting them back onto the original stack. So parentheses create new stacks and return from them. That's doing something!

Oh, before you say I'm just being pedantic, that wasn't my goal. You're 100% right w.r.t. math classrooms. I was only trying to add that the claim that "parentheses are not operators" is contextual and interpretation dependent. When adults are talking about parsing algebraic expressions, they are usually talking about compilers and syntax parsing in general, and they are usually in computing. So yeah, adding something, not taking away.

The first program I was paid to write was a simple calculator, and I implemented parenthesis as two unary do-nothing operators. It worked perfectly, and didn't add any complexity to the implementation. I don't think it's unreasonable to think of parenthesis that way, but it is a bit weird that it doesn't look for matching.

Interesting article. However, I suspect that I teach younger pupils than you do and I would add that the crossover period from single or serial arithmetic tasks into multiple operators is fraught with problems. Having a simple hook that students can hang onto is a useful tool; it gives them a flag that they can wave at relevant problems, and each other. I would heartily agree that it should not be taught as a rigid system or a blind process. Understanding how something works and the limitations of a method are important skills and I would always teach them too.

Great post, I too find this frustrating, and have first hand experience of people teaching it wrong! I wrote this recently in a similar vein: https://cavmaths.wordpress.com/2014/06/29/aaargh-ruddy-bidmas/

I just ran it to this post via another website. I am glad to see an article about alternatives to PEMDAS. For several years, I stopped using PEMDAS in my sixth grade classroom and their misconceptions like adding before all subtraction, went away completely. Well, until one of the kids came to class having his older brother help him with his homework. Suddenly he was making so many mistakes. I asked him what his brother did to "help" him and he answered "PEMDAS." I recently got all the other teachers at my school onboard with dropping PEMDAS and our scores on order of operations went up tremendously. I didn't know anyone else used GEMA. Again, like you said, it really isn't necessary to use any mnemonic at all, but it was the only way to get some to drop PEMDAS. Lovely article!

Excellent article! I'd like to see you extend the examples to include division and multiplication as well. It may be difficult to deal with division as something that's simple to understand as shorthand for something. 6 / 2 is short hand for 6 * 1/2 but students certainly are not used to that view. And the notion of left to right is important to include as a 3rd thing to remember since subtraction and division are not commutative.